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\title{MEKF}
\author{Makani Technologies LLC}
\date{October 2016\; (DRAFT)}

\begin{document}
\maketitle

\section{States}

\begin{equation}
  C_g^b \approx \hat C_g^b (I + [\psi_{gb}^g]_{\times})
\end{equation}

\begin{equation}
  \vec{b} \approx \hat{b} + \delta \vec{b}
\end{equation}

\section{Propagate}

\begin{equation}
  \delta \vec{\theta} = (\vec{\omega} - \vec{b}) \cdot \Delta t
\end{equation}

\begin{equation}
  \delta q = q(\delta \vec{\theta})
\end{equation}

\begin{equation}
  {q_g^b}_{k|k-1} = {q_g^b}_{k-1|k-1} \star \delta q
\end{equation}


\begin{equation}
  \delta \vec{x} = [\delta \vec{\theta}, \delta \vec{b}]^T
\end{equation}

\begin{equation}
  \mathbf{F} = \begin{bmatrix}
    \mathbf{C}(\delta q) & \Delta t \cdot \mathbf{I} \\
    \mathbf{0}           & \mathbf{I} \\
  \end{bmatrix}
\end{equation}

\begin{equation}
  \mathbf{\hat{x}}_{k+1|k} = \mathbf{F} \mathbf{\hat{x}}_{k|k} +
                             \mathbf{B} \mathbf{w}_{k}
\end{equation}

\begin{equation}
  \mathbf{P} = \mathbf{U} \mathbf{D} \mathbf{U}^T
\end{equation}

\begin{equation}
  \mathbf{F}_k = \frac{\partial \mathbf{f}}{\partial \mathbf{x}} \bigg|_{\mathbf{\hat{x}}_{k-1|k-1}}
\end{equation}

\begin{eqnarray}
  P_{k+1|k} &=& F_k P_{k|k} F_k^T + Q_k \\
            &=& F_k U_{k|k} D_{k|k} U_{k|k}^T F_k^T + B Q_k B^T \\
            &=& W_{k|k} D_{k|k} W_{k|k}^T
\end{eqnarray}


\begin{equation}
  \mathbf{D}_w = \begin{bmatrix}
    \mathbf{D}_{k|k} & \mathbf{0} \\
    \mathbf{0}       & \mathbf{D}_q \\
  \end{bmatrix}
\end{equation}

\begin{equation}
  \mathbf{W} = [\mathbf{F} \mathbf{U}, \mathbf{B}]
\end{equation}


\section{Correct}

\begin{equation}
  h(\mathbf{x}) \equiv C_g^b \vec{v}_g
                \approx \hat{C}_g^b (\mathbf{I} + \delta \vec{\theta}_{\times}) \vec{v}_g
\end{equation}

\begin{equation}
  \frac{\partial h}{\partial \delta \vec{\theta}} = -[\hat{C}_g^b \vec{v}_g]_{\times}
\end{equation}

\end{document}
